synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
Given a differentiable manifold $X$, the cotangent bundle $T^*(X)$ of $X$ is the dual vector bundle over $X$ dual to the tangent bundle $T x$ of $X$.
A cotangent vector or covector on $X$ is an element of $T^*(X)$. The cotangent space of $X$ at a point $a$ is the fiber $T^*_a(X)$ of $T^*(X)$ over $a$; it is a vector space. A covector field on $X$ is a section of $T^*(X)$. (More generally, a differential form on $X$ is a section of the exterior algebra of $T^*(X)$; a covector field is a differential 1-form.)
Given a covector $\omega$ at $a$ and a tangent vector $v$ at $a$, the pairing $\langle{\omega,v}\rangle$ is a scalar (a real number, usually). This (with some details about linearity and universality) is basically what it means for $T^*(X)$ to be the dual vector bundle to $T_*(X)$. More globally, given a covector field $\omega$ and a tangent vector field $v$, the paring $\langle{\omega,v}\rangle$ is a scalar function on $X$.
Given a point $a$ in $X$ and a differentiable (real-valued) partial function $f$ defined near $a$, the differential $\mathrm{d}_a f$ of $f$ at $a$ is a covector on $X$ at $a$; given a tangent vector $v$ at $a$, the pairing is given by
thinking of $v$ as a derivation on differentiable functions defined near $a$. (It is really the germ at $a$ of $f$ that matters here.) More globally, given a differentiable function $f$, the de Rham differential $\mathrm{d}f$ of $f$ is a covector field on $X$; given a vector field $v$, the pairing is given by
thinking of $v$ as a derivation on differentiable functions.
One can also define covectors at $a$ to be germs of differentiable functions at $a$, modulo the equivalence relation that $\mathrm{d}_a f = \mathrm{d}_a g$ if $f - g$ is constant on some neighbourhood of $a$. In general, a covector field won't be of the form $\mathrm{d}f$, but it will be a sum of terms of the form $h \mathrm{d}f$. More specifically, a covector field $\omega$ on a coordinate patch can be written
in local coordinates $(x^1,\ldots,x^n)$. This fact can also be used as the basis of a definition of the cotangent bundle.
Every cotangent bundle $T^\ast X$ carries itself a canonical differential 1-form
with the property that under the isomorphism
between differential 1-forms and smooth sections of the cotangent bundle we have for every smooth section $\sigma \in \Gamma(T^* X)$ the identification
between the pullback of $\theta$ along $\sigma$ and the 1-form corresponding to $\sigma$ under $j$.
This unique differential 1-form $\theta \in \Omega^1(T^* X)$ is called the Liouville-Poincaré 1-form or canonical form or tautological form on the cotangent bundle.
The de Rham differential $\omega \coloneqq d \theta$ is a symplectic form. Hence every cotangent bundle is canonically a symplectic manifold.
On a coordinate chart $\mathbb{R}^n$ of $X$ with canonical coordinate functions denoted $(x^i)$, the cotangent bundle over the chart is $T^\ast \mathbb{R}^n \simeq \mathbb{R}^n \times \mathbb{R}^n$ with canonical coordinates $((x^i), (p_j))$. In these coordinates the canonical 1-form is (using Einstein summation convention)
and hence the symplectic form is
microsupport?, microlocal sheaf theory